First, we note that the underlying family of prestable curve \(C \rightarrow S\)  is given by a morphism \(S \rightarrow \textbf {M}\)  to the stack of prestable curves over \(Spec\text{ }{k}\) , which by [1]} Appendix \(A.3\)  is an algebraic stack admitting a log structure given by the basic log structure. Using this, we can equip \(C\)  and \(S\)  with log structures \(\mathcal {M}^{pre}_C\)  and \(\mathcal {M}_S\)  such that \(C \rightarrow S\)  is log smooth. Since \(f: C \rightarrow X\)  intersects the boundary at finitely many points, the pullback log structure \( f^*\mathcal {M}_X\)  has a uniquely determined map to the divisorial log structure on the section divisor of \(C\)  which intersects \(D \subset X\) , which we denote by \(\mathcal {M}_C^{div}\) . We now equip \(C\)  with the log structure \(\mathcal {M}_C = \mathcal {M}^{pre}_C \oplus _{\mathcal {O}_C^\times } \mathcal {M}_C^{div}\) . With these log structures, \(C \rightarrow S\)  is log smooth, and we have a log lift \(C \rightarrow X\)  of the underlying morphism of schemes. Uniqueness of a log lift follows from an easy case of [2]} Theorem \(4.13\) .
